By solving the systems of linear equations given by substitution, the riddle solved is: HE MADE A KNOT IN HIS TAIL AND CALLED IT A PIG'S TIE
To solve the riddle, the following systems of linear equations would be solved by finding the values of x and y by substitution:
A. 3x + 2y = 12 --> Eqn. 1
y = x - 9 --> Eqn. 2
Substitute y = x - 9 into eqn. 1 to find x
Substitute x = 6 into eqn. 2 to find y
The solution is (6, -3) which tallies with the word: HE
B. 4x + y = -2 --> Eqn. 1
y = 2x - 2 --> Eqn. 2
Substitute y = 2x - 2 into eqn. 1 to find x
Substitute x = 0 into eqn. 2 to find y
The solution is (0, -2) which tallies with the word: MADE
C. -3x + 5y = 5 --> Eqn. 1
y = x - 1 --> Eqn. 2
Substitute y = x - 1 into eqn. 1 to find x
Substitute x = 5 into eqn. 2 to find y
The solution is (5, 4) which tallies with the word: A
D. 2x + y = -16 --> Eqn. 1
y = 2x --> Eqn. 2
Substitute y = 2x into eqn. 1 to find x
Substitute x = -4 into eqn. 2 to find y
The solution is (-4, -8) which tallies with the word: KNOT
E. 7x = -35 --> Eqn. 1
-8x + 9y = 4 --> Eqn. 2
Solve eqn. 1. to find x
Substitute x = -5 into eqn. 2 to find y
The solution is (-5, -4) which tallies with the word: IN
F. -4x + 3y = 20 --> Eqn. 1
-14y = -56 --> Eqn. 2
Solve eqn. 2 to find y
Substitute y = 4 into eqn. 1 to find y
The solution is (-2, 4) which tallies with the word: HIS
G. 13x - 6y = -5 --> Eqn. 1
x + 10 = 11 --> Eqn. 2
Solve eqn. 2 to find x
x = 11 - 10
x = 1
Substitute x = 1 into eqn. 1 to find y
The solution is (1, 3) which tallies with the word: TAIL
i. x = 6 + 2y --> Eqn. 1
-3x + 14y = -18 --> Eqn. 2
Substitute x = 6 + 2y into eqn. 2 to find y
Substitute y = 0 into eqn. 1 to find x
The solution is (6, 0) which tallies with the word: CALLED
H. 9x - 2y = 12 --> Eqn. 1
y + 4 = 16 --> Eqn. 2
Solve eqn. 2 to find y
y = 16 - 4
y = 12
Substitute y = 12 into eqn. 1 to find x
The solution is (4, 12) which tallies with the word: AND
I. x = 6 + 2y --> Eqn. 1
-3x + 14y = -18 --> Eqn. 2
Substitute x = 6 + 2y into eqn. 2 to find y
Substitute y = 0 into eqn. 1 to find x
The solution is (6, 0) which tallies with the word: CALLED
J. 5x - 9y = 12 --> Eqn. 1
y = -6 - x --> Eqn. 2
Substitute y = -6 - x into eqn. 1 to find x
Substitute x = -3 into eqn. 2 to find y
The solution is (-3, -3) which tallies with the word: IT
K. x = -8 + y--> Eqn. 1
6x + y = -6 --> Eqn. 2
Substitute x = -8 + y into eqn. 2 to find y
Substitute y = 6 into eqn. 1 to find x
x = -8 + 6
x = -2
The solution is (-2, 6) which tallies with the word: A
L. 7x - 3y = 17 --> Eqn. 1
y = 2x - 6 --> Eqn. 2
Substitute y = 2x - 6 into eqn. 1 to find x
Substitute x = -1 into eqn. 2 to find y
The solution is (-1, -8) which tallies with the word: PIGS
M. Given that the total number of push-ups and pull-ups equals 36,
Let,
x = number of push-upsy = number of pull-ups
The first equation that can be formed from this is:
x + y = 36 --> Eqn. 1
Also, if the instructor wants 8 times as many push-ups as pull-ups from the students, we would have the second equation as:
y = 8x --> Eqn. 2
Substitute y = 8x into eqn. 1 to find the value of x
Number of push-ups in 1 min = 4
Substitute x = 4 into eqn. 2 to find the value of y
Number of push-ups in 1 min = 32
The solution to the system of linear equations would be (4, 32) which corresponds to the word: TIE.
Therefore, by solving the systems of linear equations given by substitution, the riddle solved is: HE MADE A KNOT IN HIS TAIL AND CALLED IT A PIG'S TIE
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